Sampled-data repetitive control for a class of non-minimum phase nonlinear systems subject to period variation

ABSTRACT

The robust sampled-data repetitive control (RC, or repetitive controller, also designated RC) problem for non-minimum phase nonlinear systems is both challenging and practical. This paper proposes a sampled-data output-feedback RC design for a class of non-minimum phase systems with measurable nonlinearities to improve the robustness against the period variation. The design relies on additive-state decomposition, by which the output-feedback RC problem is decomposed into an output-feedback RC problem for a linear time-invariant component and a state-feedback stabilisation problem for a nonlinear component. Thanks to the decomposition, existing controller design methods in both the frequency domain and the time domain are employed to make the robustness and discretisation for a continuous-time nonlinear system tractable. In order to demonstrate the effectiveness, an illustrative example is given.

KEYWORDS:

• Repetitive control
• sampled-data systems
• non-minimum phase
• nonlinear systems
• robustness
• additive state decomposition

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The neutral-type system in the critical case is in the form of $\dot{x}\left(t\right)$ $-\dot{x}(t-T)$ $=A_{1}x\left(t\right)$ $+A_{2}x(t-T),$ where $x\left(t\right)\in {\mathbb{R}}^n,$ $T>0,$ and $A_1,A_2\in {\mathbb{R}}^{n\times n}$ (Hale & Verduyn, 1993; Quan et al., 2010).

2 A continuous-time controller is first designed based on a continuous-time plant model. The sampling is completely ignored at this step. Then, the obtained continuous-time controller is discretised and implemented using a sample-and-hold device (Nesić & Teel, 2001).

3 By (9(9) $\begin{array}{rl}{\hat{y}}_p& =y-c^{\mathrm{T}}{\hat{x}}_s\end{array}$(9) ) and (10(10) $\begin{array}{rl}{\dot{\hat{x}}}_s& =A{\hat{x}}_s+b{u}_s+\phi \left(y\right)-\phi \left(r\right),{\hat{x}}_s\left(0\right)=0.\end{array}$(10) ), ${\hat{y}}_p$ and ${\hat{x}}_s$are obtained instead of the true state x. So far, x or $x_p$ is still unknown. In fact, we avoid solving x by using the additive-state decomposition.

4 Here, $\mathcal{B}\left(\delta \right)\triangleq \{\xi \in \mathbb{R}|\parallel \xi \parallel \le \delta \},$ $\delta >0;$ the notation $x\left(t\right)\to \mathcal{B}\left(\delta \right)$ means $\underset{y\in \mathcal{B}\left(\delta \right)}{min}$ $\left|x\right(t)-y|\to 0$as $t\to \mathrm{\infty }.$

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 61973015] and Beijing Natural Science Foundation [grant number L182037].

Notes on contributors

Quan Quan

Quan Quan received the B.S. and Ph.D. degrees from Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China, in 2004, and 2010, respectively. He has been an Associate Professor with the School of Automation Science and Electrical Engineering, Beihang University since 2013. His research interests include reliable flight control, vision based navigation, repetitive learning control, and time-delay systems.

Kai-Yuan Cai

Kai-Yuan Cai received the B.S., M.S., and Ph.D. degrees from Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China, in 1984, 1987, and 1991, respectively. He has been a full Professor at Beihang University since 1995. His main research interests include software testing, software reliability, reliable flight control, and software cybernetics. Dr. Cai is a Cheung Kong Scholar (Chair Professor), jointly appointed by the Ministry of Education of China and the Li Ka Shing Foundation of Hong Kong in 1999.